A kitchen supply company manufactures two conical funnels, Funnel P and Funnel Q, which both have a height of 24...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A kitchen supply company manufactures two conical funnels, Funnel P and Funnel Q, which both have a height of 24 centimeters. The radius of the circular opening of Funnel P is 10 centimeters. The radius of the circular opening of Funnel Q is 20% smaller than the radius of Funnel P. What is the volume, in cubic centimeters, of Funnel Q? (The volume of a cone with radius r and height h is \(\mathrm{V = \frac{1}{3}\pi r^2h}\).)
- \(\mathrm{340\pi}\)
- \(\mathrm{512\pi}\)
- \(\mathrm{640\pi}\)
- \(\mathrm{800\pi}\)
1. TRANSLATE the problem information
- Given information:
- Both funnels have height \(\mathrm{h = 24\text{ cm}}\)
- Funnel P has radius \(\mathrm{r_P = 10\text{ cm}}\)
- Funnel Q's radius is "20% smaller" than Funnel P's radius
- Volume formula: \(\mathrm{V = \frac{1}{3}\pi r^2h}\)
- What "20% smaller" means: If something is 20% smaller, it retains 80% of its original size
2. TRANSLATE to find Funnel Q's radius
- Since Funnel Q is 20% smaller: \(\mathrm{r_Q = r_P \times 0.8}\)
- \(\mathrm{r_Q = 10\text{ cm} \times 0.8 = 8\text{ cm}}\)
3. SIMPLIFY the volume calculation
- Apply the cone volume formula: \(\mathrm{V_Q = \frac{1}{3}\pi r_Q^2h}\)
- Substitute known values: \(\mathrm{V_Q = \frac{1}{3}\pi(8)^2(24)}\)
- \(\mathrm{V_Q = \frac{1}{3}\pi(64)(24)}\)
- \(\mathrm{V_Q = \frac{1}{3}\pi(1,536)}\)
- \(\mathrm{V_Q = 512\pi\text{ cubic centimeters}}\)
Answer: B. 512π
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "20% smaller" and calculate \(\mathrm{r_Q = 10 \times 0.2 = 2\text{ cm}}\) instead of \(\mathrm{r_Q = 10 \times 0.8 = 8\text{ cm}}\).
When they use \(\mathrm{r = 2\text{ cm}}\) in the volume formula: \(\mathrm{V = \frac{1}{3}\pi(2)^2(24) = \frac{1}{3}\pi(4)(24) = 32\pi}\). This doesn't match any answer choice exactly, leading to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly find \(\mathrm{r_Q = 8\text{ cm}}\) but make arithmetic errors during volume calculation, such as computing \(\mathrm{(8)^2 = 16}\) instead of \(\mathrm{64}\), or incorrectly handling the fraction \(\mathrm{\frac{1}{3}}\).
This may lead them to select Choice A (340π) if they make multiple calculation errors.
The Bottom Line:
This problem tests whether students can correctly interpret percentage decrease language and then execute multi-step calculations accurately. The key insight is recognizing that "20% smaller" means keeping 80% of the original value, not multiplying by 20%.